Parabola


Def: Parabola is the locus of a point on a plane, which is
equidistant from a fixed point and a fixed line of that plane.
If P is the variable
point, S is the fixed
point and PM is the distance
from a fixed line 'd', then PS = PM (FocusDirectrix
property)
Standard form: y^{2} = 4ax, y^{2} =  4ax, x^{2} = 4ay, x^{2} =  4ay
Important information:




y^{2}
= 4ax

y^{2}
=  4ax

x^{2}
= 4ay

x^{2}
=  4ay

Coordinate of Vertex

(0, 0)

(0, 0)

(0, 0)

(0, 0)

Coordinate of focus

(a, 0)

(
a, 0)

(0, a)

(0,  a)

Axis

xaxis

xaxis

yaxis

yaxis

Equation of directrix

x + a = 0

x  a = 0

y + a = 0

y  a = 0

Length of Latus rectum

4a

4a

4a

4a

Coordinate of the end
of Latus rectum

(a, 2a)
(a, 
2a)

(
a, 2a)
(
a, 
2a)

(2a, a)
( 2a, a)

(2a, 
a)
( 2a,  a)

Parametric Coordinate

x = at^{2}, y =
2at

x = 
at^{2}, y = 2at

x = 2at , Y = at^{2}

x = 2at, y =  at^{2}

Equation of axis

y = 0

y = 0

x = 0

x = 0

Extent(Quadrants)

I & IV

II & III

I & II

III & IV




General Equation: (y  b)^{2} = 4a(x  a)






Important information:



Coordinate of Vertex

Coordinate of focus

Equation of
directrix

Length of Latus
rectum

Coordinate of
the end of Latus rectum

Equation of axis

Parametric Co

(a, b)

(a + a,b)

X = a + a

4a

(a + a, 2a + b) (a + a,  2a + b)

y = b

x = at^{2 }+ a,
y = 2at + b







Tangent to a parabola:



Equation of tangent to a parabola y^{2} = 4ax at (x_{1}, y_{1}) is yy_{1} = 2a(x + x_{1})
Equation of tangent to a parabola y^{2} = 4ax at (at^{2},2at) is yt = x + at^{2}.



General equation of tangent to a parabola y^{2} = 4ax in terms of
slope is y = mx +

a


m





General equation of tangent to y^{2} = 4ax in terms of parameter 't' is yt = x + at^{2}.



Condition of tangency c =

a


m





Point of contact is

(




a


m^{2}


,

a


m


)






Ellipse:


Def: Ellipse is the locus of a point on a plane such
that the ratio e of its distances
from a fixed point and a fixed line of that plane is constant where e <1.
If P is the variable
point, S is the fixed
point and PM is the distance
from a fixed line 'd' then PS = e PM (FocusDirectrix property)






Standard form:

x^{2}


a^{2}


+


y^{2}


b^{2}


= 1(a > b)







x^{2}


a^{2}


+


y^{2}


b^{2}


= 1(a > b)



Coordinate of
Vertex

(0, 0)

Coordinate of
focus

(± ae, 0)

Major axis

xaxis

Length of Major
axis

2a

Minor axis

yaxis

Length of Minor
axis

2b

Equation of
directrix

x ±

a


e


= 0


Length of Latus
rectum

2b^{2}


a


Coordinate of
the end of Latus rectum

(

± ae, ±

b^{2}


a


)


Relation
between a, b and e

b^{2} = a^{2}(1  e^{2})

Parametric
Coordinate

x = a cosq, y = b sinq







Tangent to an Ellipse:



Equation of tangent to an ellipse

x^{2}


a^{2}


+

y^{2}


b^{2}


= 1 (a > b) at (x^{1}, y^{1}) is

xx_{1}


a^{2}


+

yy_{1}


b^{2}


= 1

Equation of tangent to an ellipse

x^{2}


a^{2}


+

y^{2}


b^{2}


= 1 at (a cosq, b sinq) is

xcosq


a


+

ysinq


b


= 1

General equation of tangent to an ellipse

x^{2}


a^{2}


+

y^{2}


b^{2}


= 1 in terms of slope m is


y = mx


Condition of tangency c =


Point of contact is

(

±



m


,


)



Hyperbola:


Def: Hyperbola is the
locus of a point on a plane such that the ratio e of its distances
from a fixed point and a fixed line of that plane is constant where e >1.
If P is the variable
point, S is the fixed
point and PM is the distance
from a fixed line 'd' then PS = e PM (FocusDirectrix
property)
Standard form:

x^{2}


a^{2}





y^{2}


b^{2}


= 1(a > b)







Important information:



x^{2}


a^{2}





y^{2}


b^{2}


= 1(a > b)



Coordinate of
Centre

(0,0)

Coordinate of
focus

(± ae, 0)

Major axis

xaxis

Length of Major
axis

2a

Minor axis

yaxis

Length of Minor
axis

2b

Equation of
directrix

X ±

a


e


= 0


Length of Latus
rectum

2b^{2}


a


Relation
between a,b and e

b^{2} = a^{2}(e^{2}  1 )

Coordinate of
the end of Latus rectum

(

± ae, ±

b^{2}


a


)


Parametric
Coordinate

X = a secq, y = b tanq







Tangent to a Hyperbola



Equation of tangent to a hyperbola

x^{2}


a^{2}




y^{2}


b^{2}


= 1 (a > b) at (x1, y1) is

xx_{1}


a^{2}




yy_{1}


b^{2}


= 1

Equation of tangent to a hyperbola

x^{2}


a^{2}




y^{2}


b^{2}


= 1 at (a secq, b tanq) is

x secq


a


+

y tanq


b


= 1

General equation of tangent to a hyperbola

x^{2}


a^{2}




y^{2}


b^{2}


= 1 in terms of slope m is


y = mx


Condition of tangency c =


Point of contact is

(




m


,


)




